Chapter 11

For Problems 1-22, find the partial fraction decomposition for each rational expression. See answers below. \(\frac{11 x-10}{(x-2)(x+1)}\)

For Problems \(1-32\), use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{rl}2 x-y & =-2 \\ 3 x+2 y & =11\end{array}\right)\)

For Problems \(1-12\), evaluate each \(2 \times 2\) determinant by using Definition 11.1. \(\left|\begin{array}{ll}4 & 3 \\ 2 & 7\end{array}\right|\)

For Problems \(1-10\), indicate whether each matrix is in reduced echelon form. \(\left[\begin{array}{ll:r}1 & 0 & -4 \\ 0 & 1 & 14\end{array}\right]\)

Find the partial fraction decomposition for each rational expression. See answers below. \(\frac{11 x-2}{(x+3)(x-4)}\)

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{c}3 x+y=-9 \\ 4 x-3 y=1\end{array}\right)\)

Evaluate each \(2 \times 2\) determinant by using Definition 11.1. \(\left|\begin{array}{ll}3 & 5 \\ 6 & 4\end{array}\right|\)

Indicate whether each matrix is in reduced echelon form. \(\left[\begin{array}{ll:l}1 & 2 & 8 \\ 0 & 0 & 0\end{array}\right]\)

Use the graphing approach to determine whether the system is consistent, the system in inconsistent, or the equations are dependent. If the system is consistent, find the solution set from the graph and check it. \(\left(\begin{array}{rl}3 x+y & =0 \\ x-2 y & =-7\end{array}\right)\)

Find the partial fraction decomposition for each rational expression. See answers below. \(\frac{-2 x-8}{x^{2}-1}\)

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{l}5 x+2 y=5 \\ 3 x-4 y=29\end{array}\right)\)

Evaluate each \(2 \times 2\) determinant by using Definition 11.1. \(\left|\begin{array}{rr}-3 & 2 \\ 7 & 5\end{array}\right|\)

Indicate whether each matrix is in reduced echelon form. \(\left[\begin{array}{lll:l}1 & 0 & 2 & 5 \\ 0 & 1 & 3 & 7 \\ 0 & 0 & 0 & 0\end{array}\right]\)

Use the graphing approach to determine whether the system is consistent, the system in inconsistent, or the equations are dependent. If the system is consistent, find the solution set from the graph and check it. \(\left(\begin{array}{l}4 x+3 y=-5 \\ 2 x-3 y=-7\end{array}\right)\)

Find the partial fraction decomposition for each rational expression. See answers below. \(\frac{-2 x+32}{x^{2}-4}\)

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{l}4 x-7 y=-23 \\ 2 x+5 y=-3\end{array}\right)\)

Evaluate each \(2 \times 2\) determinant by using Definition 11.1. \(\left|\begin{array}{rr}5 & 3 \\ 6 & -1\end{array}\right|\)

Indicate whether each matrix is in reduced echelon form. \(\left[\begin{array}{lll:r}1 & 0 & 0 & 5 \\ 0 & 3 & 0 & 8 \\ 0 & 0 & 1 & -11\end{array}\right]\)

Use the graphing approach to determine whether the system is consistent, the system in inconsistent, or the equations are dependent. If the system is consistent, find the solution set from the graph and check it. \(\left(\begin{array}{l}2 x-y=9 \\ 4 x-2 y=11\end{array}\right)\)

Find the partial fraction decomposition for each rational expression. See answers below. \(\frac{20 x-3}{6 x^{2}+7 x-3}\)

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{c}5 x-4 y=14 \\ -x+2 y=-4\end{array}\right)\)

Evaluate each \(2 \times 2\) determinant by using Definition 11.1. \(\left|\begin{array}{ll}2 & -3 \\ 8 & -2\end{array}\right|\)

Indicate whether each matrix is in reduced echelon form. \(\left[\begin{array}{rrr:r}1 & 0 & 0 & 17 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & -14\end{array}\right]\)

Use the graphing approach to determine whether the system is consistent, the system in inconsistent, or the equations are dependent. If the system is consistent, find the solution set from the graph and check it. \(\left(\begin{array}{l}\frac{1}{2} x+\frac{1}{4} y=9 \\ 4 x+2 y=72\end{array}\right)\)

Find the partial fraction decomposition for each rational expression. See answers below. \(\frac{-2 x-8}{10 x^{2}-x-2}\)

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{rl}-x+2 y & =10 \\ 3 x-y & =-10\end{array}\right)\)

Indicate whether each matrix is in reduced echelon form. \(\left[\begin{array}{rrr:r}1 & 0 & 0 & -7 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 9\end{array}\right]\)

Use the graphing approach to determine whether the system is consistent, the system in inconsistent, or the equations are dependent. If the system is consistent, find the solution set from the graph and check it. \(\left(\begin{array}{lr}5 x+2 y= & -9 \\ 4 x-3 y= & 2\end{array}\right)\)

Find the partial fraction decomposition for each rational expression. See answers below. \(\frac{x^{2}-18 x+5}{(x-1)(x+2)(x-3)}\)

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{l}y=2 x-4 \\ 6 x-3 y=1\end{array}\right)\)

Evaluate each \(2 \times 2\) determinant by using Definition 11.1. \(\left|\begin{array}{ll}-2 & -3 \\ -1 & -4\end{array}\right|\)

Indicate whether each matrix is in reduced echelon form. \(\left[\begin{array}{rrr:r}1 & 1 & 0 & -3 \\ 0 & 1 & 2 & 5 \\ 0 & 0 & 1 & 7\end{array}\right]\)

Use the graphing approach to determine whether the system is consistent, the system in inconsistent, or the equations are dependent. If the system is consistent, find the solution set from the graph and check it. \(\left(\begin{array}{l}\frac{1}{2} x-\frac{1}{3} y=3 \\ x+4 y=-8\end{array}\right)\)

Find the partial fraction decomposition for each rational expression. See answers below. \(\frac{-9 x^{2}+7 x-4}{x^{3}-3 x^{2}-4 x}\)

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{l}-3 x-4 y=14 \\ -2 x+3 y=-19\end{array}\right)\)

Evaluate each \(2 \times 2\) determinant by using Definition 11.1. \(\left|\begin{array}{ll}-4 & -3 \\ -5 & -7\end{array}\right|\)

Indicate whether each matrix is in reduced echelon form. \(\left[\begin{array}{rrr:r}1 & 0 & 3 & 8 \\ 0 & 1 & 2 & -6 \\ 0 & 0 & 0 & 0\end{array}\right]\)

Use the graphing approach to determine whether the system is consistent, the system in inconsistent, or the equations are dependent. If the system is consistent, find the solution set from the graph and check it. \(\left(\begin{array}{l}4 x-9 y=-60 \\ \frac{1}{3} x-\frac{3}{4} y=-5\end{array}\right)\)

Find the partial fraction decomposition for each rational expression. See answers below. \(\frac{-6 x^{2}+7 x+1}{x(2 x-1)(4 x+1)}\)

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{rl}-4 x+3 y & =3 \\ 4 x-6 y & =-5\end{array}\right)\)

Indicate whether each matrix is in reduced echelon form. \(\left[\begin{array}{rrrr:r}1 & 0 & 0 & 3 & 4 \\ 0 & 1 & 0 & 5 & -3 \\ 0 & 0 & 1 & -1 & 7 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]\)

Use the graphing approach to determine whether the system is consistent, the system in inconsistent, or the equations are dependent. If the system is consistent, find the solution set from the graph and check it. \(\left(\begin{array}{r}x-\frac{y}{2}=-4 \\ 8 x-4 y=-1\end{array}\right)\)

Find the partial fraction decomposition for each rational expression. See answers below. \(\frac{15 x^{2}+20 x+30}{(x+3)(3 x+2)(2 x+3)}\)

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{l}x=4 y-1 \\ 2 x-8 y=-2\end{array}\right)\)

Evaluate each \(2 \times 2\) determinant by using Definition 11.1. \(\left|\begin{array}{ll}\frac{2}{3} & \frac{3}{4} \\ 8 & 6\end{array}\right|\)

Indicate whether each matrix is in reduced echelon form. \(\left[\begin{array}{llll:r}1 & 0 & 0 & 0 & 2 \\ 0 & 0 & 1 & 0 & 4 \\ 0 & 1 & 0 & 0 & -3 \\ 0 & 0 & 0 & 1 & 9\end{array}\right]\)

Use the graphing approach to determine whether the system is consistent, the system in inconsistent, or the equations are dependent. If the system is consistent, find the solution set from the graph and check it. \(\left(\begin{array}{l}3 x-2 y=7 \\ 6 x+5 y=-4\end{array}\right)\)

Find the partial fraction decomposition for each rational expression. See answers below. \(\frac{2 x+1}{(x-2)^{2}}\)

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{l}9 x-y=-2 \\ 8 x+y=4\end{array}\right)\)

For Problems \(11-30\), use a matrix approach to solve each system. \(\left(\begin{array}{rl}x-3 y & =14 \\ 3 x+2 y & =-13\end{array}\right)\)